CN102497337B - Compressed sensing wireless communication channel estimation method based on sparsity self-adapting - Google Patents

Abstract

The invention belongs to the field of wireless communication channel estimation, particularly relates to a compressed sensing wireless communication channel estimation method based on sparsity self-adapting, which includes the ssteps: (1) collecting demodulated receiving signals and calculating channel response of a pilot frequency position; (2) constructing a measurement matrix phi required by signal reconstruction; (3) calculating an association degree vector and sequencing elements of the vector; (4) calculating second difference vector of a novel association degree vector after sequencing and setting a threshold value I for judging sparsity of signals; (5) estimating sparsity S of channel impulse response; (6) comparing the threshold value I with the last element of a vector D sequentially, and a coefficient value corresponding to the first element larger than the threshold value is the estimated sparsity S of the signals; and (7) reconstructing the signals. The channel estimation method breaks a bottleneck of a traditional compressed sensing algorithm that the sparsity of the signals must be known, and signal reconstruction of sparsity self-adapting is achieved.

Description

A kind of based on the adaptive compressed sensing radio communication channel of degree of rarefication method of estimation

Affiliated technical field

The invention belongs to radio communication channel and estimate field, particularly for the multicarrier condition of sparse channel under dual-selection channel condition, estimate.

Background technology

Compressed sensing (Compressive Sensing, CS) theory is a quantum jump of applied mathematics and signal process field, its represent when signal be compressible or when certain transform domain has sparse property, by gathering a small amount of signal projection, just can realize the accurate or approximate reconstruct of signal.Under this theoretical frame, sampling rate is no longer decided by the bandwidth of signal, but is decided by structure and the content of information in signal, thereby has broken the Bottleneck Restrictions of traditional nyquist sampling theorem to sample rate.Compressive sensing theory, sampling and the compression of signal can be carried out with low rate simultaneously, sample frequency and data storage and the transmission cost of signal have greatly been reduced, reduced significantly signal processing time and assessed the cost, thereby the proposition of compressed sensing is a major transformation of signal process field.

Multi-transceiver technology utilizes a series of orthogonal sub-carriers to realize the high-speed transfer of data, it is a kind of efficient parallel data transmission plan, wherein OFDM (OFDM, Orthogonal Frequency Division Multiplexing) technology is most widely used multi-transceiver technology.The main feature of OFDM is that high-speed serial data is divided into the parallel transmission that carries out relative low speed in a plurality of orthogonal sub-carriers.Owing to having orthogonality between each subcarrier, allow the frequency spectrum of subchannel overlapped, thereby the OFDM availability of frequency spectrum is higher.In addition, the anti-frequency selective fading performance of OFDM technology is strong, realizes simply, easily eliminates intersymbol interference.Quadrature modulation in each sub-channels of OFDM and demodulation can adopt IFFT and FFT method to realize, and greatly reduce the complexity of calculating.Yet ofdm system is very responsive to phase noise and carrier wave frequency deviation, and because the frequency spectrum of subchannel covers mutually, this has just proposed strict requirement to the orthogonality between subcarrier.Because wireless transmission channel characteristic is undesirable, conventionally present time domain and the decline of frequency domain double selectivity, very easily cause system frequency difference.The existence of system frequency difference, by destroying the orthogonality of ofdm system sub-carriers, produces inter-carrier interference (ICI), severe exacerbation systematic function.Thereby, to the accurate estimation of channel, be to guarantee that ofdm system possesses the key of premium properties.

In wireless OFDM communication system, the channel estimation method based on pilot tone is topmost channel estimating means.For multicarrier system, pilot tone generally has time-frequency two-dimensional characteristic, therefore need to apply two-dimension pilot frequency method of estimation.Two-dimension pilot frequency method of estimation generally comprises two steps: (1) estimating pilot frequency time of living in or frequency location place channel response, its mathematical optimization criterion used comprises least square (LS) algorithm, the least mean-square error estimation technique (Minimum Mean Square Error, MMSE), maximum likelihood estimate (Maximum Likehood, ML).(2), on the basis of channel response that obtains pilot tone position, by certain two-dimentional interpolation method, obtain the estimation to complete channel response.Two-dimensional interpolation can be decomposed into the one dimension interpolation of two cascades conventionally, and main one dimension interpolation method comprises: linear (Linear) interpolation, Gauss interpolation, Cubic interpolation, Lagrange's interpolation and DFT interpolation etc.The compound mode of conventional two-dimensional interpolation mainly comprises Linear-DFT two-dimensional interpolation and DFT-DFT two-dimensional interpolation.

Yet there is following defect in traditional two-dimensional interpolation technology: in actual transmission of wireless signals, the multipath channel of double selectivity is conventionally only dominated by a predominating path bunch institute for minority, so the physical channel presenting often has sparse characteristic.And when the transmission bandwidth of signal is large or antenna number is more, the sparse characteristic of channel is particularly evident.Because condition of sparse channel only has minority non-zero tap, traditional method based on pilot frequency sequence very likely samples the zero tap of channel, and interpolation goes out channel response exactly.And compressed sensing technology can fully be excavated the sparse characteristic of channel, can utilize very limited pilot tone effectively to recover sparse channel impulse response.At present, existing scholar both domestic and external is applied to existing CS classic algorithm to going in the estimation of condition of sparse channel, but because existing most of CS algorithm all needs the degree of rarefication of known signal as the prerequisite of signal reconstruction, this is difficult to realize in actual applications, therefore, need to there is novel degree of rarefication self-adapting reconstruction algorithm, can be the in the situation that of the unknown of signal degree of rarefication, still can accurately recover signal, realize the accurate estimation to condition of sparse channel.

Summary of the invention

Under doubly-selective fading channel condition, conventional channel estimation technique cannot accurately be estimated the problem of condition of sparse channel, the present invention proposes a kind of multicarrier system channel estimation methods that can reduce pilot number, improve the availability of frequency spectrum of wireless communication system.The channel estimation methods that the present invention proposes, without the degree of rarefication of known channel impulse response, has been broken through the bottleneck of the necessary known signal degree of rarefication of conventional compression perception algorithm, has realized the adaptive signal reconstruction of degree of rarefication.Technical scheme of the present invention is as follows:

Based on the adaptive compressed sensing radio communication channel of a degree of rarefication method of estimation, comprise the following steps:

1) at receiving terminal, gather the reception signal after demodulation, by it divided by pilot tone amplitude, to calculate the channel response H at pilot tone place _p, and using it as recovering the required measurement vector of whole channel response;

2) the required measurement matrix Φ of structure signal reconstruction;

3) compute associations degree vector Φ ^*h _p, and all elements in this vector is pressed to the descending sequence of amplitude, the new degree of association vector after being sorted;

4) calculate the second differnce vector D of the new degree of association vector after sequence, and according to the average amplitude of rear 50% element of vectorial D, be provided for the threshold value I of decision signal degree of rarefication;

5) estimate the degree of rarefication S of channel impulse response: from last element of vectorial D, compare successively with set threshold value I, first corresponding coefficient value of element that is greater than threshold value is estimated signal degree of rarefication S;

6) carry out signal reconstruction: to measure vectorial H _pas residual error r _tinitial value r ₀, select interconnection vector Φ ^*r _tthe S of a middle amplitude maximum element, and the corresponding coefficient of this S element is saved in to a minute quantity set Γ _tin, wherein t is iteration pointer, is used to indicate iterations, initial value is 0;

7) by the up-to-date minute quantity set Γ identifying _twith the current minute quantity set F approaching _t-1merge, obtain intersection U _t, current approximation component collection F wherein _tinitial condition be empty;

8) according to the intersection U after upgrading _tspecified coefficient is selected measures corresponding row in matrix Φ, and compute associations vector Φ again ^*r, selects the wherein element of S amplitude maximum, by the current collection F that approaches _tbe updated to the corresponding coefficient of this S element;

9) calculate new residual values: wherein, for row coefficient belongs to a minute quantity set F _tthe submatrix of measurement matrix Φ, for pseudo inverse matrix;

10) judge whether residual values is less than preset value, if can not meet, iteration pointer t adds 1, and returns to the 6th step and repeat above step, until meet stopping criterion for iteration;

11) if iterations reaches, still cannot to make residual values meet after the upper limit pre-conditioned, according to formula the estimated value of revision degree of rarefication, and make the zero clearing of iteration pointer, return to the 6th step, until that residual values meets is pre-conditioned, accurately reconstruct channel impulse response, wherein, bracket function in ceil () expression, the span of η is (1,2], n is used to refer to generation revision number of times, often once revises, and n value adds 1.

The present invention can be according to the equidistant condition of following satisfied constraint, the required measurement matrix Φ of structure signal reconstruction: for any c and constant δ _k∈ (0,1), measuring matrix Φ need meet $(1-{\mathrm{\δ}}_K){\left|\right|c\left|\right|}_2^2\≤{\left|\right|{\mathrm{\Φ}}_{T}c\left|\right|}_2^2\≤(1+{\mathrm{\δ}}_K){\left|\right|c\left|\right|}_2^2,$ Wherein, index for the set of index number, c is sparse signal, is an one-dimensional vector that length is identical with the dimension of T, and the degree of rarefication of establishing sparse signal c is S, Φ _tfor measuring the submatrix of the M * T consisting of the indicated related column of index T in matrix Φ, integer M and N are respectively line number and the columns of measuring matrix Φ

Conventional channel estimation technique, owing to cannot excavating the sparse characteristic of channel, in channel life period and frequency double selectivity decline situation, cannot estimate channel response exactly.Compressed sensing technology of the present invention can make full use of the sparse characteristic of double selectivity fade condition lower channel, and the sampled point (being the channel response at pilot tone place) of utilization minute quantity just can recover the impulse response of channel integral body.Due to the minimizing of required sampled point, the required pilot-frequency expense of system also will significantly reduce, and therefore, the present invention will contribute to improve wireless communication system, especially the availability of frequency spectrum of broad band multicarrier system.Meanwhile, what the present invention adopted is novel degree of rarefication self-adapting compressing perception algorithm, and with respect to traditional compressed sensing algorithm that signal degree of rarefication is known that requires, the method just can accurately be recovered original signal without known signal degree of rarefication.This characteristic makes the present invention possess stronger practical value.

Accompanying drawing explanation

Fig. 1 is FB(flow block) of the present invention.

Fig. 2 is for estimating the flow chart of channel degree of rarefication.

Fig. 3 is channel impulse response reconstruct flow chart.

Embodiment

As shown in Figure 1, the present invention is mainly divided into three steps: obtain pilot tone place channel response, estimate channel impulse response degree of rarefication and reconstruct channel impulse response.

Concrete scheme is as follows:

One, pilot tone place channel impulse response obtains

If the time-domain and frequency-domain discrete representation of channel transfer function H (f, t) is H _{l, k}, l=0 ..., L-1, k=0 ..., K-1, the subcarrier number that wherein L is each multicarrier symbol, K is the symbol numbers that every frame comprises.Frequency pilot sign is expressed as N in the spacing of frequency direction _f, in the spacing of time orientation, be expressed as N _t.

The reception signal of a multicarrier frame is so:

R _l，k＝H _l，kS _l，k+Z _l，k(l＝0，...，L-1，k＝0，...，K-1) (1)

R wherein _{l, k}for the receiving symbol after demodulation, S _{l, k}for sending symbol, Z _{l, k}for Gaussian noise, and time-frequency discrete channel coefficient H _{l, k}portrayed an equivalent system channel, comprising multi-carrier modulator, interpolation filter, physical channel and frequency overlapped-resistable filter.Therefore, H _{l, k}can be expressed as:

$H_{l.k}=\underset{m=0}{\overset{K-1}{\mathrm{\Σ}}}\underset{i=-L/2}{\overset{L/2-1}{\mathrm{\Σ}}}F[m,i]e^{-j2\mathrm{\π}(\frac{\mathrm{km}}{K}-\frac{\mathrm{li}}{L})}---\left(2\right)$

Wherein,

$F[m,i]=\underset{q=0}{\overset{T-1}{\mathrm{\Σ}}}{S}_h[m,i+\mathrm{qL}]A_{\mathrm{\γ},g}^{*}(m,\frac{i+\mathrm{qL}}{\mathrm{LT}})---\left(3\right)$

And represent Discrete Time-Delay-Doppler distribution function, T is-symbol duration, h[n, m] become impulse response while being discrete; for mutual ambiguity function.

All frequency pilot signs in one frame can be expressed as set P, and the number of frequency pilot sign is:

$N_{\mathrm{frid}}=\left[\frac{L}{N_f}\right]\left[\frac{K}{N_t}\right]=\left|\right|P\left|\right|---\left(4\right)$

Therefore, the channel coefficients of insertion frequency pilot sign is:

$H_P={\hat{H}}_{l^{\′},k^{\′}}=\frac{R_{l^{\′},k^{\′}}}{S_{l^{\′},k^{\′}}}=H_{n^{\′},i^{\′}}+\frac{Z_{l^{\′},k^{\′}}}{S_{l^{\′},k^{\′}}},\∀(l^{\′},k^{\′})\∈P---\left(5\right)$

Two, the estimation of channel degree of rarefication

(1) Wireless Channel Modeling based on compressed sensing

In actual wireless communication transmissions process, because communication environments is comparatively complicated, multipath phenomenon is obvious, and most wireless channel is actually only dominated by a few path cluster, and therefore, physical channel often presents sparse property.This characteristic can be embodied by Discrete Time-Delay-Doppler distribution function, i.e. S _h[m, i] is that S is sparse or " compressible ".In condition of sparse channel, S _h[m, i] only has S coefficient to keep off in zero, and this provides prerequisite for applied compression cognition technology.

If compressive sensing theory thinks that the one-dimensional signal x that a length is N is that S is sparse, and meets S much smaller than N, as long as known some M * N (M < N) dimension is measured matrix Φ, and the linear measurement value y of x under this matrix,

y＝Φx (6)

Just can from measured value y, recover original signal x.Because M is much smaller than N, so compressed sensing technology only need utilize the sampled value of minute quantity can reconstruct signal.Yet measure matrix, the equidistant condition of constraint must be met, Accurate Reconstruction could be realized, for any S sparse signal c and constant δ _k∈ (0,1), measures matrix Φ and meets

$(1-{\mathrm{\&delta;}}_K){\left|\right|c\left|\right|}_2^2\&le;{\left|\right|{\mathrm{\&Phi;}}_{T}c\left|\right|}_2^2\&le;(1+{\mathrm{\&delta;}}_K){\left|\right|c\left|\right|}_2^2,\&ForAll;c\&Element;R^{\left|\right|T\left|\right|}---\left(7\right)$

Wherein, || T|| represents the dimension of T, and the dimension of T is less than S, and c is the one-dimensional vector that any one length is identical with T dimension, Φ _tsubmatrix for M * T of being formed by the indicated related column of index T in Φ.

Due to S _h[m, i] possesses sparse property, thereby the F[m in (3) formula, i] possess too sparse property.Discrete channel model can be converted into the compressed sensing reconstruction model shown in (6) formula.(2) formula is reduced to:

$H_{\mathrm{\&lambda;},\mathrm{\&kappa;}}=\underset{m=0}{\overset{K-1}{\mathrm{\&Sigma;}}}\underset{i=-L/2}{\overset{L/2-1}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{m,i}{u}_{m,i}[\mathrm{\&lambda;},\mathrm{\&kappa;}]---\left(8\right)$

Wherein,

${\mathrm{\&alpha;}}_{m,i}=\sqrt{\mathrm{LK}}F[m,i]---\left(9\right)$

$u_{m,i}[\mathrm{\&lambda;},\mathrm{\&kappa;}]=(1/\sqrt{\mathrm{LK}})e^{-j2\mathrm{\&pi;}(\mathrm{\&kappa;m}/D-\mathrm{\&lambda;i}/j)}---\left(10\right)$

Variable λ ∈ in above formula (0, L-1), κ ∈ (0, K-1), H _{λ, κ}and u _{m, i}[λ, κ] is L * K matrix.The dimensional vector h=vec{H that definition length is LK _{λ, κ}, the element in h is by H _{λ, κ}in column vector be connected and form successively.In like manner can define,

u _m，i＝vec{u _m，i[λ，κ]} (11)

So, (8) formula can be rewritten as

$h=\underset{m=0}{\overset{K-1}{\mathrm{\&Sigma;}}}\underset{i=-L/2}{\overset{L/2-1}{\mathrm{\&Sigma;}}}{\mathrm{\&alpha;}}_{m,i}{u}_{m,i}=\mathrm{\&Psi;\&alpha;}---\left(12\right)$

Wherein, α=vec{ α _{m, i}, Ψ is LK * LK matrix, its ((i+L/2) K+m+1) row are vectorial u _{m, i}.Due to vectorial u _{m, i}for orthogonal vectors, so matrix Ψ meets constraint isometry.

If H _pfor discrete channel coefficient H _{l, k}at (l, k), belong to the corresponding channel response in pilot set P place, Φ is in the specified Ψ of set P || P|| row form || and P|| * LK matrix.(12) formula can be converted into the compressed sensing model in (6) formula so:

H _p＝Φα (13)

Wherein, we have obtained the channel impulse response H at pilot tone place _p, Φ is the measurement matrix in this model, just can recover α by (13) formula, thereby obtain the impulse response of overall channel.

(2) the channel degree of rarefication estimation technique based on second differnce

The first step of reconstruction signal α is to identify in α to have which " atom " (being the element in α) to participate in the measurement of signal.Atom is higher with the degree of association of measuring matrix, more likely participates in measuring.The degree of association of atom characterizes as it is at vectorial Φ ^*h _pthe amplitude of middle corresponding element, therefore vectorial Φ ^*h _palso referred to as degree of association vector.Atom and the degree of association of measuring matrix also can be described as the energy of atom, and due to the energy of the atom that participates in the measuring energy much larger than other atoms, thereby the scope of its energy hunting is also larger.And the atom that has neither part nor lot in measurement is because energy own is less, its energy hunting scope is also less.Therefore, by observing the speed of nuclear energy decline, just can distinguish total which atom participation measurement, i.e. degree of rarefication of signal.

As shown in Figure 2, the channel degree of rarefication estimation technique based on second differnce, needs first compute associations vector Φ ^*h _p, and its element is pressed to the descending sequence of amplitude, the speed that nuclear energy declines can be by Φ ^*h _psecond differnce characterize,

D＝diff ₂(Φ ^*H _p) (14)

Choose Φ ^*h _pthe average amplitude of rear 50% element is as with reference to threshold value, and is multiplied by coefficient δ and obtains threshold value I,

I＝δ·ave(|D(0.5*(LK-2):LK-2)|) (15)

Wherein, ave (| the average amplitude (length of D is LK-2) of 50% element after D (0.5* (LK-2): LK-2) |) representation vector D.

From last element of D, compare successively with threshold value I, first corresponding coefficient of element that is greater than I is the degree of rarefication of channel.

Three, reconstruct channel impulse response

After estimating channel degree of rarefication, the iterative algorithm that returns order renewal by utilization can reconstruct original signal (as shown in Figure 3).Specific algorithm is as follows:

1. initialization residual error r _t, minute quantity set Γ t, maximum tolerance residual values ε, iterations upper limit t _maxand the current collection F that approaches _t.Wherein t is iteration pointer, often carry out an iteration t value and increase 1; Residual error r _tinitial value for measuring vectorial H _p; Divide quantity set Γ t and the current collection F that approaches _tinitial value be all empty set.

2. calculate a minute quantity set Γ t.The new residual error r that utilizes last iteration to produce _t-1, and calculate new degree of association vector Φ ^*rt-1, and therefrom identify the element of S amplitude maximum, and by it at Φ ^*in rt-1, corresponding position deposits in minute quantity set Γ t.

3. merge minute quantity set Γ t and the resulting current collection F that approaches in last iteration _t-1, obtain intersection U _t.

4. according to intersection U _tthe indicated row structure of middle element is measured the submatrix Φ of matrix Φ _ut, and again differentiate the element of middle S amplitude maximum ( for matrix pseudo inverse matrix), and F _tin content update be the corresponding position parameter of this S element.

5. according to the current collection F that approaches _tindicated row structure is measured the submatrix Φ of matrix Φ _ut, and recalculate residual error

6. if residual error is less than maximum tolerance residual values ε, iteration finishes, and reconstruct completes; Otherwise, return to the 2nd step and continue iteration.

7. if iterations reaches the upper limit, and residual error do not meet pre-conditionedly yet, revises estimated channel degree of rarefication, and revisal formulas is $S_n=\mathrm{ceil}({\mathrm{\&eta;}}^{{(-1)}^n+\mathrm{ceil}(n/2)}\&times;S).$ Wherein, bracket function in ceil () expression, the span of η be (1,2], n is used to refer to revision number of times.Along with the increase of n, due to continuous the increase and positive-negative polarity checker of index amplitude of η, so correction value S _nto constantly depart from S to both direction.The large young pathbreaker of η determines the size of the each skew of correction value.Simulation result shows, because estimated value extremely approaches actual value, therefore, only needs to revise several times and can realize Accurate Reconstruction through minority.

Complete after above-mentioned steps, can reconstruct vectorial α, and then recover F[m, i].According to formula (2), just can calculate again the time-domain and frequency-domain discrete representation H of whole channel _{l, k}thereby, realize condition of sparse channel and estimate.

A specific embodiment of the present invention below:

1. adopt OFDM modulation system, and with Rayleigh 5 footpath channels for channel to be estimated, its degree of rarefication is 30.Sub-carrier number is 128, and the symbolic number of each subcarrier carrying is 12.Time domain direction pilot interval is 4, and frequency domain direction pilot interval is 4, so pilot tone number is 96, and pilot-frequency expense is only 6.25%.

2. use (8) to (12) formula to calculate measurement matrix, and set up the channel estimation model based on compressed sensing.The channel degree of rarefication estimation technique of employing based on second differnce estimated channel degree of rarefication, and coefficient δ is made as 7, and the estimated value calculating is 32.

3. by the restructing algorithm in the estimated channel degree of rarefication substitution three going out in 2, wherein maximum tolerance residual values ε is made as 10 ^-4, maximum iteration time is made as 25, η value and is made as 1.2.Through 18 iteration, realize signal reconstruction.

Claims (2)

1. based on the adaptive compressed sensing radio communication channel of a degree of rarefication method of estimation, comprise the following steps:

1) at receiving terminal, gather the reception signal after demodulation, by it divided by pilot tone amplitude, to calculate the channel response H at pilot tone place _p, and using it as recovering the required measurement vector of whole channel response;

2) the required measurement matrix Φ of structure signal reconstruction;

3) compute associations degree vector Φ * H _p, and all elements in this vector is pressed to the descending sequence of amplitude, the new degree of association vector after being sorted;

4) calculate the second differnce vector D of the new degree of association vector after sequence, and according to the average amplitude of rear 50% element of vectorial D, be provided for the threshold value I of decision signal degree of rarefication;

5) estimate the degree of rarefication S of channel impulse response: from last element of vectorial D, compare successively with set threshold value I, first corresponding coefficient value of element that is greater than threshold value is estimated signal degree of rarefication S;

6) carry out signal reconstruction: to measure vectorial H _pas residual error r _tinitial value r ₀, select interconnection vector Φ * r _tthe S of a middle amplitude maximum element, and the corresponding coefficient of this S element is saved in to a minute quantity set Γ _tin, wherein t is iteration pointer, is used to indicate iterations, initial value is 0;

7) by the up-to-date minute quantity set Γ identifying _twith the current minute quantity set F approaching _t-1merge, obtain intersection U _t, the current minute quantity set F approaching wherein _tinitial condition be empty;

8) according to the intersection U after upgrading _tspecified coefficient is selected measures corresponding row in matrix Φ, and compute associations vector Φ * r again _t, select the wherein element of S amplitude maximum, by the current minute quantity set F approaching _tbe updated to the corresponding coefficient of this S element;

9) calculate new residual values: wherein, for row coefficient belongs to a minute quantity set F _tthe submatrix of measurement matrix Φ, for pseudo inverse matrix;

10) judge whether residual values is less than preset value, if can not meet, iteration pointer t adds 1, and returns to the 6th step and repeat above step, until meet stopping criterion for iteration;

11) if iterations reaches, still cannot to make residual values meet after the upper limit pre-conditioned, according to formula the estimated value of revision degree of rarefication, and make the zero clearing of iteration pointer, return to the 6th step, until that residual values meets is pre-conditioned, accurately reconstruct channel impulse response, wherein, bracket function in ceil () expression, the span of η is (1,2], n is used to refer to generation revision number of times, often once revises, and n value adds 1.

2. according to claim 1 based on the adaptive compressed sensing radio communication channel of degree of rarefication method of estimation, it is characterized in that, step 2) in, according to the equidistant condition of following satisfied constraint, the required measurement matrix Φ of structure signal reconstruction: for any c and constant δ _k∈ (0,1), measuring matrix Φ need meet wherein, index for the set of index number, c is sparse signal, is an one-dimensional vector that length is identical with the dimension of T, and the degree of rarefication of establishing sparse signal c is S, Φ _tfor measuring the submatrix of the M * T consisting of the indicated related column of index T in matrix Φ, integer M and N are respectively line number and the columns of measuring matrix Φ.